Algebra Examples

$\frac{15 d^{- 9} t^{4}}{{(5 d^{- 3} t)}^{- 2}}$

Step 1

Simplify.

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Step 1.1

Move $d^{- 9}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{15 t^{4}}{{(5 d^{- 3} t)}^{- 2} d^{9}}$

Step 1.2

Move ${(5 d^{- 3} t)}^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{15 t^{4} {(5 d^{- 3} t)}^{2}}{d^{9}}$

Step 2

Simplify the numerator.

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Step 2.1

Apply the product rule to $5 d^{- 3} t$ .

$\frac{15 t^{4} {(5 d^{- 3})}^{2} t^{2}}{d^{9}}$

Step 2.2

Multiply $t^{4}$ by $t^{2}$ by adding the exponents.

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Step 2.2.1

Move $t^{2}$ .

$\frac{15 (t^{2} t^{4}) {(5 d^{- 3})}^{2}}{d^{9}}$

Step 2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{15 t^{2 + 4} {(5 d^{- 3})}^{2}}{d^{9}}$

Step 2.2.3

Add $2$ and $4$ .

$\frac{15 t^{6} {(5 d^{- 3})}^{2}}{d^{9}}$

Step 2.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{15 t^{6} {(5 \frac{1}{d^{3}})}^{2}}{d^{9}}$

Step 2.4

Combine $5$ and $\frac{1}{d^{3}}$ .

$\frac{15 t^{6} {(\frac{5}{d^{3}})}^{2}}{d^{9}}$

Step 2.5

Apply the product rule to $\frac{5}{d^{3}}$ .

$\frac{15 t^{6} \frac{5^{2}}{{(d^{3})}^{2}}}{d^{9}}$

Step 2.6

Raise $5$ to the power of $2$ .

$\frac{15 t^{6} \frac{25}{{(d^{3})}^{2}}}{d^{9}}$

Step 2.7

Multiply the exponents in ${(d^{3})}^{2}$ .

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Step 2.7.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{15 t^{6} \frac{25}{d^{3 \cdot 2}}}{d^{9}}$

Step 2.7.2

Multiply $3$ by $2$ .

$\frac{15 t^{6} \frac{25}{d^{6}}}{d^{9}}$

Step 2.8

Combine exponents.

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Step 2.8.1

Combine $15$ and $\frac{25}{d^{6}}$ .

$\frac{t^{6} \frac{15 \cdot 25}{d^{6}}}{d^{9}}$

Step 2.8.2

Multiply $15$ by $25$ .

$\frac{t^{6} \frac{375}{d^{6}}}{d^{9}}$

Step 2.8.3

Combine $t^{6}$ and $\frac{375}{d^{6}}$ .

$\frac{\frac{t^{6} \cdot 375}{d^{6}}}{d^{9}}$

Step 2.9

Move $375$ to the left of $t^{6}$ .

$\frac{\frac{375 t^{6}}{d^{6}}}{d^{9}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{375 t^{6}}{d^{6}} \cdot \frac{1}{d^{9}}$

Step 4

Combine.

$\frac{375 t^{6} \cdot 1}{d^{6} d^{9}}$

Step 5

Multiply $d^{6}$ by $d^{9}$ by adding the exponents.

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Step 5.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{375 t^{6} \cdot 1}{d^{6 + 9}}$

Step 5.2

Add $6$ and $9$ .

$\frac{375 t^{6} \cdot 1}{d^{15}}$

Step 6

Multiply $375 t^{6}$ by $1$ .

$\frac{375 t^{6}}{d^{15}}$